A Giffen good is an extreme type of inferior good. The negative income effect of changes in price of a Giffen good is actual stronger than the substitution effect. This leads to its bizarre quality: when the price of a Giffen good rises, consumers actually buy more. Veblen goods behave the same way for very different reasons.
To understand how this happens, consider the example of a Giffen good for which there is the best evidence that it is a Giffen good. Households in the Hunan province of China were shown to buy more rice when they had to buy it at a higher price, and less when the price they paid was subsidised.
The reason for this is that, even when expensive, rice was still the cheapest source of calories available. Therefore, when the price of rice was cut, households had more money left over after buying rice. Some of this was spent on buying more expensive foods (meat, vegetables and fruit), which reduced their need for rice.
For many years the classic example of a Giffen good were potatoes during the Irish potato famine. One problem with this was partly a lack of data to prove it was a Giffen good. more fundamentally, given that the price rises were caused by lower supply, how could people have been consuming more? This example is mentioned here merely because it still occurs in many text-books — although it is likely to be dropped from future editions now that the Chinese example is known!
Giffen goods may be very rare, but the fact that some exist is important: the complete lack of any examples of a phenomenon that economic theory said was possible, was, to quote the paper on the Hunanese rice, “an embarrassment to economists“